General Electric synchrotron accelerator built in 1946, the origin of the discovery of synchrotron radiation. The arrow indicates the evidence of radiation.

Synchrotron radiation was named after its discovery in a General Electric synchrotron accelerator built in 1946 and announced in May 1947 by Frank Elder, Anatole Gurewitsch, Robert Langmuir, and Herb Pollock in a letter entitled "Radiation from Electrons in a Synchrotron".[1] Pollock recounts:

"On April 24, Langmuir and I were running the machine and as usual were trying to push the electron gun and its associated pulse transformer to the limit. Some intermittent sparking had occurred and we asked the technician to observe with a mirror around the protective concrete wall. He immediately signaled to turn off the synchrotron as "he saw an arc in the tube." The vacuum was still excellent, so Langmuir and I came to the end of the wall and observed. At first we thought it might be due to Cherenkov radiation, but it soon became clearer that we were seeing Ivanenko and Pomeranchuk radiation."[2]

When high-energy particles are in rapid motion, including electrons forced to travel in a curved path by a magnetic field, synchrotron radiation is produced. This is similar to a radio antenna, but with the difference that, in theory, the relativistic speed will change the observed frequency due to the Doppler effect by the Lorentz factor, . Relativistic length contraction then bumps the frequency observed in the lab by another factor of , thus multiplying the GHz frequency of the resonant cavity that accelerates the electrons into the X-ray range. The radiated power is given by the relativistic Larmor formula while the force on the emitting electron is given by the Abraham–Lorentz–Dirac force. The radiation pattern can be distorted from an isotropic dipole pattern into an extremely forward-pointing cone of radiation. Synchrotron radiation is the brightest artificial source of X-rays. The planar acceleration geometry appears to make the radiation linearly polarized when observed in the orbital plane, and circularly polarized when observed at a small angle to that plane. Amplitude and frequency are however focussed to the polar ecliptic.

Synchrotron radiation may occur in accelerators either as a nuisance, causing undesired energy loss in particle physics contexts, or as a deliberately produced radiation source for numerous laboratory applications. Electrons are accelerated to high speeds in several stages to achieve a final energy that is typically in the GeV range. In the LHC proton bunches also produce the radiation at increasing amplitude and frequency as they accelerate with respect to the vacuum field, propagating photoelectrons, which in turn propagate secondary electrons from the pipe walls with increasing frequency and density up to 7x1010. Each proton may lose 6.7keV per turn due to this phenomenon.[3]

Messier 87's Energetic Jet, HST image. The blue light from the jet emerging from the bright AGN core, towards the lower right, is due to synchrotron radiation.

Synchrotron radiation is also generated by astronomical objects, typically where relativistic electrons spiral (and hence change velocity) through magnetic fields. Two of its characteristics include non-thermal power-law spectra, and polarization.[4]

Crab Nebula. The bluish glow from the central region of the nebula is due to synchrotron radiation.

Supermassive black holes have been suggested for producing synchrotron radiation, by ejection of jets produced by gravitationally accelerating ions through the super contorted 'tubular' polar areas of magnetic fields. Such jets, the nearest being in Messier 87, have been confirmed by the Hubble telescope as apparently superluminal, travelling at 6×c (six times the speed of light) from our planetary frame. This phenomenon is caused because the jets are travelling very near the speed of light and at a very small angle towards the observer. Because at every point of their path the high-velocity jets are emitting light, the light they emit does not approach the observer much more quickly than the jet itself. Light emitted over hundreds of years of travel thus arrives at the observer over a much smaller time period (ten or twenty years) giving the illusion of faster than light travel. There is no violation of special relativity.[8]

A class of astronomical sources where synchrotron emission is important is the pulsar wind nebulae, a.k.a. plerions, of which the Crab nebula and its associated pulsar are archetypal. Pulsed emission gamma-ray radiation from the Crab has recently been observed up to ≥25 GeV,[9] probably due to synchrotron emission by electrons trapped in the strong magnetic field around the pulsar. Polarization in the Crab[10] at energies from 0.1 to 1.0 MeV illustrates a typical synchrotron radiation.

which is the unit vector between the observation point and the position of the charge at the retarded time, and is the retarded time.

In equation (1), and (2), the first terms fall off as the inverse square of the distance from the particle, and this first term is called the generalized Coulomb field or velocity field. And the second terms fall off as the inverse first power of the distance from the source, and it is called the radiation field or acceleration field. If we ignore the velocity field, the radial component of Poynting's Vector resulted from the Liénard–Wiechert field can be calculated to be

Note that

The spatial relationship between and determines the detailed angular power distribution.

The relativistic effect of transforming from the rest frame of the particle to the observer's frame manifests itself by the presence of the factors in the denominator of Eq. (3).

When the electron velocity approaches the speed of light, the emission pattern is sharply collimated forward.

When the charge is in instantaneous circular motion, its acceleration is perpendicular to its velocity . Choosing a coordinate system such that instantaneously is in the z direction and is in the x direction, with the polar and azimuth angles and defining the direction of observation, the general formula Eq. (4) reduces to

In the relativistic limit , the angular distribution can be written approximately as

The factors in the denominators tip the angular distribution forward into a narrow cone like the beam of a headlight pointing ahead of the particle. A plot of the angular distribution (dP/dΩ vs. γθ) shows a sharp peak around θ=0.

Integration over the whole solid angle yields the total power radiated by one electron

where E is the electron energy, B is the magnetic field, and ρ is the radius of curvature of the track in the field. Note that the radiated power is proportional to , , and . In some cases the surfaces of vacuum chambers hit by synchrotron radiation have to be cooled because of the high power of the radiation.

The energy received by an observer (per unit solid angle at the source) is

Using the Fourier Transformation we move to the frequency space

Angular and frequency distribution of the energy received by an observer (consider only the radiation field)

Therefore, if we know the particle's motion, cross products term, and phase factor, we could calculate the radiation integral. However, calculations are generally quite lengthy (even for simple cases as for the radiation emitted by an electron in a bending magnet, they require Airy function or the modified Bessel functions).

For frequencies much larger than the critical frequency and angles much larger than the critical angle, the synchrotron radiation emission is negligible.

Integrating on all angles, we get the frequency distribution of the energy radiated.

Frequency distribution of radiated energy

If we define

, where . Then,

Note that , if , and , if

The formula for spectral distribution of synchrotron radiation, given above, can be expressed in terms of a rapidly converging integral with no special functions involved [12] (see also modified Bessel functions ) by means of the relation:

Then, the relationship between radiated power and photon energy is shown in the graph on the right side. The higher the critical energy, the more photons with high energies are generated. Note that, there is no dependence on the energy at longer wavelength.

In Eq.(10), the first term is the radiation power with polarization in the orbit plane, and the second term is the polarization orthogonal to the orbit plane. In the orbit plane , the polarization is purely horizontal. Integrating on all frequencies, we get the angular distribution of the energy radiated

Integrating on all the angles, we find that seven times as much energy is radiated with parallel polarization as with perpendicular polarization. The radiation from a relativistically moving charge is very strongly, but not completely, polarized in the plane of motion.